What is a natural SUSY scenario?

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Published for SISSA by Springer Received: September 8, 2014 Revised: April 17, 2015 Accepted: May 12, 2015 Published: June 11, 2015

What is a natural SUSY scenario?

J. Alberto Casas,^a Jes´us M. Moreno,^a Sandra Robles,^a,c Krzysztof Rolbiecki^a and Bryan Zald´ıvar^b

aInstituto de F´ısica Te´orica, IFT-UAM/CSIC, Universidad Aut´onoma de Madrid, Cantoblanco, 28049 Madrid, Spain

bService de Physique Th´eorique, Universit´e Libre de Bruxelles, Boulevard du Triomphe, CP225, 1050 Brussels, Belgium

cDepartamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain

E-mail: alberto.casas@uam.es,jesus.moreno@csic.es, sandra.robles@uam.es,krzysztof.rolbiecki@desy.de, bryan.zaldivar@ulb.ac.be

Abstract: The idea of “Natural SUSY”, understood as a supersymmetric scenario where the fine-tuning is as mild as possible, is a reasonable guide to explore supersymmetric phe- nomenology. In this paper, we re-examine this issue in the context of the MSSM including several improvements, such as the mixing of the fine-tuning conditions for different soft terms and the presence of potential extra fine-tunings that must be combined with the electroweak one. We give tables and plots that allow to easily evaluate the fine-tuning and the corresponding naturalness bounds for any theoretical model defined at any high-energy (HE) scale. Then, we analyze in detail the complete fine-tuning bounds for the uncon- strained MSSM, defined at any HE scale. We show that Natural SUSY does not demand light stops. Actually, an average stop mass below 800 GeV is disfavored, though one of the stops might be very light. Regarding phenomenology, the most stringent upper bound from naturalness is the one on the gluino mass, which typically sets the present level fine-tuning at O(1%). However, this result presents a strong dependence on the HE scale. E.g. if the latter is 10⁷GeV the level of fine-tuning is ∼ four times less severe. Finally, the most robust result of Natural SUSY is by far that Higgsinos should be rather light, certainly below 700 GeV for a fine-tuning of O(1%) or milder. Incidentally, this upper bound is not far from ' 1 TeV, which is the value required if dark matter is made of Higgsinos.

Keywords: Supersymmetry Phenomenology ArXiv ePrint: 1407.6966

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Contents

1 Introduction 1

2 The Natural SUSY scenario. A critical review 3

2.1 The “standard” Natural SUSY 3

2.2 The dependence on the initial parameters 4

2.3 Fine-tunings left aside 6

3 The electroweak fine-tuning of the MSSM 7

3.1 The measure of the fine-tuning 8

3.2 Generic expression for the fine-tuning 10

3.3 The fit to the low-energy quantities 12

4 The naturalness bounds 13

4.1 Bounds on the initial (high-energy) parameters 13

4.2 Correlations between the soft terms 16

4.3 Bounds on the supersymmetric spectrum 17

5 Results for the unconstrained MSSM 19

6 Impact of other potential fine-tunings of the MSSM 21

6.1 Fine-tuning to get m^exp_h ' 125 GeV 21

6.2 The Higgs mass and the parameter space selected by naturalness 23

6.3 Fine-tuning to get large tan β 24

7 Summary and conclusions: the most robust predictions of a Natural

SUSY scenario 26

A Low-energy running coefficients at 2 loops 29

1 Introduction

The idea of “Natural SUSY” has become very popular in the last times, especially as a framework that justifies that e.g. the stops should be light (much lighter than the other squarks), say m˜t<

∼600 GeV. This is an attractive scenario since it gives theoretical support to searches for light stops and other particles at the LHC, a hot subject from the theoretical and the experimental points of view.

In a few words, the idea is to lie in a region of the minimal supersymmetric Standard Model (MSSM) parameter-space where the electroweak breaking is not fine-tuned (or not

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too much fine-tuned). This is reasonable since, as it is usually argued, the main phe- nomenological virtue of supersymmetry (SUSY) is precisely to avoid the huge fine-tuning associated to the hierarchy problem.

Then, the main argument is in brief the following: “Stops produce the main radiative contributions to the Higgs potential, in particular to the Higgs mass-parameter m². To avoid fine-tunings these contributions should be reasonably small, not much larger than m² itself. Since they are proportional to the stop masses, the latter cannot be too large”.

Other supersymmetric particles, like gluinos, are also constrained by the same reason; in particular the gluino mass is bounded from above due to its important contribution to the running of the stop masses, which implies a significant 2-loop contribution to m². In addition, Higgsinos should be light, as their masses are controlled by the µ−parameter, which contributes to m². These statements sound reasonable and have been often used to quantify the “naturalness” upper bounds on stop masses, gluino masses, etc. Apart from the theoretical arguments, the experiments at the LHC, ATLAS and CMS, performed a large number of SUSY searches, covering a significant part of the parameter space [1–8].

From the point of view of Natural SUSY models, the most interesting bounds are those for stops, gluinos and Higgsinos. The lower limits from direct production of stops reach as far as 650 GeV [9], but they are sensitive to the stop details, in particular the mass difference between the stop and the lightest supersymmetric particle (LSP). Much lighter stops are still allowed by the current experimental bounds once certain conditions on their decays are fulfilled [10–12]. Concerning the gluino, the current experimental bounds have a strong dependence on the masses of the light squarks. Assuming that the stops are the only squarks lighter than the gluino (as suggested by the very “Natural SUSY” rationale), the latter decays through a chain ˜g→ t¯t˜χ⁰₁, and the lower limit reaches mg˜ <

∼ 1.4 TeV [13].

Again, with some additional assumptions on the decay chains this limit can be somewhat relaxed. Finally, the µ parameter is the least constrained at the LHC. Because of the low electroweak production cross-section and the large model dependence, it is entirely possible to have Higgsinos just above the current LEP limit, µ & 95 GeV [14, 15]. On the other hand, the LEP limit is rather model independent, even if the Higgsino is the LSP with three almost mass degenerate states around 100 GeV. All these bounds are relevant to establish the present degree of fine-tuning in different SUSY scenarios.

Another important experimental ingredient in connection with Natural SUSY is the physics of the Higgs boson [16,17]. In particular, the Higgs mass plays a prominent role in naturalness arguments. According to the most recent analyses, m_h = 125.36±0.41 GeV [18]

(ATLAS) and mh = 125.03± 0.30 GeV [19] (CMS). It is interesting to note that the current measurements are already more accurate than the theoretical predictions, which have a ∼ 2 − 3 GeV uncertainty [20–22]. Furthermore, all observed Higgs properties are remarkably close to the SM predictions [23], which, within the SUSY context, points to the decoupling limit [24].

In this paper, we revisit the arguments leading to the previous Natural SUSY sce- nario, showing that some of them are weak or incomplete. In section 2, we review the

“standard” Natural SUSY scenario, pointing out some weaknesses in the usual evaluation of its electroweak fine-tuning, i.e. the tuning to get the correct electroweak scale. We also

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address the existence of two potential extra fine-tunings that cannot be ignored in the discussion, namely the tuning to get mh = m^exp_h when stops are too light and the tuning to get a large tan β. In section 3, we discuss the electroweak fine-tuning of the MSSM, showing its statistical meaning and its generic expression for any theoretical framework.

We give in the appendix tables and plots that allow to easily evaluate the fine-tuning for any theoretical model within the framework of the MSSM, at any value of the high-energy scale. In section 4, we describe our method to rigorously extract bounds on the initial (high-energy) parameters and on the supersymmetric spectrum, from the fine-tuning con- ditions. In section 5, we apply this method to obtain the numerical values of the various naturalness bounds for the unconstrained MSSM, defined at arbitrary high-energy scale, in a systematic way. In section 6, we evaluate the impact of the potential extra fine-tunings mentioned above, discussing also the correlation between soft terms that the experimental Higgs mass imposes in the MSSM and its consequences for the electroweak fine-tuning.

Finally, in section 7 we present the summary and conclusions of the paper, outlining the main characteristics of Natural SUSY and their level of robustness against changes in the theoretical framework or the high-energy scale at which the soft parameters appear.

2 The Natural SUSY scenario. A critical review 2.1 The “standard” Natural SUSY

Naturalness arguments have been used since long ago [25] to constrain from above super- symmetric masses.¹ Already in the LHC era, they were re-visited in ref. [76] to formulate the so-called Natural SUSY scenario. For the purpose of later discussion, we summarize in this subsection the argument of ref. [76], which have been invoked in many papers.

Assuming that the extra (supersymmetric) Higgs states are heavy enough, the Higgs potential can be written in the Standard Model (SM) way

V = m²|H|²+ λ|H|⁴, (2.1)

where the SM-like Higgs doublet, H, is a linear combination of the two supersymmetric Higgs doublets, H ∼ sin βHu+ cos βHd. Then, the absence of fine-tuning can be expressed as the requirement of not-too-large contributions to the Higss mass parameter, m². Since the physical Higgs mass is m²_h = 2|m²|, a sound measure of the fine-tuning is² [51]

∆ =˜    

δm² m²

= 2δm²

m²_h . (2.2)

For large tan β, the value of m² is given by m² = |µ|²+ m²_Hu, so one immediately notes that both µ and m_Hu should be not-too-large in order to avoid fine-tuning (as has been well-known since many years ago). For the µ−parameter this implies

µ^<∼200 GeV m_h 120 GeV

 ˜∆⁻¹ 20%

!−1/2

. (2.3)

1For a partial list of references on naturalness in SUSY, see [26–57] (before LHC) and [58–75].

2This measure produces similar results to the somewhat standard parametrization of the fine-tuning, see eq. (3.4) below.

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This sets a constraint on Higgsino masses. Constraints for other particles come from the radiative corrections to m²_Hu. The most important contribution comes from the stops.

Following ref. [76]

δm²_Hu|stop =− 3

8π²y_t² m²_Q3+ m²_U3+|A^t|²
log

 Λ TeV



, (2.4)

where Λ denotes the scale of the transmission of SUSY breaking to the observable sector and the 1-loop leading-log (LL) approximation was used to integrate the renormalization-group equation (RGE). Then, the above soft parameters m²_Q3, m²_U3 and At are to be understood at low-energy, and thus they control the stop spectrum. This sets an upper bound on the stop masses. In particular one has

qm²_˜t

1+ m²_˜t

2 . 600 GeV sin β (1 + x²)^1/2

 log (Λ/ TeV) 3

−1/2 ˜∆⁻¹ 20%

!−1/2

, (2.5)

where x = A_t/q m²_˜t

1 + m²_˜t

2. Eq. (2.5) imposes a bound on the lightest stop. Besides the stops, the most important contribution to mHu is the gluino one, due to its large 1-loop RG correction to the stop masses. Again, in the 1-loop LL approximation used in ref. [76], one gets

δm²_Hu|gluino' − 2

π²y_t² αs

π



|M3|²log²

 Λ TeV



, (2.6)

where M₃ is the gluino mass. From the previous equation,

M₃. 900 GeV sin β log (Λ/ TeV) 3

−1 m_h 120 GeV

 ˜∆⁻¹ 20%

!−1/2

. (2.7)

Altogether, the summary of the minimal requirements for a natural SUSY spectrum, as given in ref. [76], is:

• two stops and one (left-handed) sbottom, both below 500 − 700 GeV.

• two Higgsinos, i.e., one chargino and two neutralinos below 200 − 350 GeV. In the absence of other [lighter] chargino/neutralinos, their spectrum is quasi-degenerate.

• a not too heavy gluino, below 900 GeV − 1.5 TeV.

In the next subsections we point out the weak points of the above arguments that support the ‘standard” Natural SUSY scenario. Part of those points have been addressed in the literature after ref. [76] (see ref. [77] for a recent and sound presentation of the naturalness issue in SUSY and references therein.)

2.2 The dependence on the initial parameters

The one-loop LL approximation used to write eqs. (2.5), (2.6), from which the naturalness bounds were obtained, is too simplistic in two different aspects.

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First, it is not accurate enough since the top Yukawa-coupling, y_t, and the strong coupling, αs, are large and vary a lot along the RG running. As a result, the soft masses evolve greatly and cannot be considered as constant, even as a rough estimate. This effect can be incorporated by integrating numerically the RGE, which corresponds to summing the leading-logs at all orders [78–81].

Second, and even more important, the physical squark, gluino and electroweakino masses are not initial parameters, but rather a low-energy consequence of the initial pa- rameters at the high-energy scale. This means that one should evaluate the cancellations required among those initial parameters in order to get the correct electroweak scale. This entails two complications. First, there is not one-to-one correspondence between the initial parameters and the physical quantities, since the former get mixed along their coupled RGEs. Consequently, it is not possible in general to determine individual upper bounds on the physical masses, not even on the initial parameters. Instead, one should expect to obtain contour-surfaces with equal degree of fine-tuning in the parameter-space and, sim- ilarly, in the “space” of the possible supersymmetric spectra. The second complication is that the results depend (sometimes critically) on what one considers as initial parameters.

The most dramatic example of the last statements is the dependence of m²_H

u on the stop masses in the constrained MSSM (CMSSM). In the CMSSM one assumes universality of scalar masses at the GUT scale, MX. This is a perfectly reasonable assumption that takes place in well-motivated theoretical scenarios, such as minimal supergravity. Then one has to evaluate the impact of the initial parameters on m²_Hu, and see whether or not the requirement of no-fine-tuning implies necessarily light stops. A most relevant analytic study concerning this issue is the well-known work by Feng et al. [54], where they studied the focus point [35,54,55] region of the CMSSM . In the generic MSSM, the (1-loop) RG evolution of a shift in the initial values of m²_Hu, m²_U

3, m²_Q

3 reads d

dt





 δm²_Hu δm²_U

3

δm²_Q

3





= y_t² 8π²





 3 3 3 2 2 2 1 1 1











 δm²_Hu δm²_U

3

δm²_Q

3





, (2.8)

where t ≡ ln Q, with Q the renormalization-scale, and yt is the top Yukawa coupling.

Hence, starting with the CMSSM universal condition at M_X: m²_Hu = m²_U

3 = m²_Q

3 = m²₀, one finds

δm²_Hu= δm²₀ 2

 3 exp

Z t 0

6y²_t 8π²dt⁰



− 1



. (2.9)

Provided tan β is large enough, exph

6 8π²

Rt 0 y²_tdt⁰i

' 1/3 for the integration between MX

and the electroweak scale, so the value of m²_H

u depends very little (in the CMSSM) on the initial scalar mass, m0. However, the average stop mass is given by (see eq. (4.21) below) m²_t˜' 2.97M3²+ 0.50m²₀+· · · , (2.10) where M3 is the gluino mass at MX. Therefore, if the stops are heavy because m0 is large, this does not imply fine-tuning. This is a clear counter-example to the need of having light stops to ensure naturalness.

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From the previous discussion it turns out that the most rigorous way to analyze the fine- tuning is to determine the full dependence of the electroweak scale (and other potentially fine-tuned quantities) on the initial parameters, and then derive the regions of constant fine- tuning in the parameter space. These regions can be (non-trivially) translated into constant fine-tuning regions in the space of possible physical spectra. This goal is enormously simplified if one determines in the first place the analytical dependence of low-energy quantities on the high-energy initial parameters, a task which will be carefully addressed in subsection3.3.

2.3 Fine-tunings left aside

In a MSSM scenario, there are two implicit potential fine-tunings that have to be taken into account to evaluate the global degree of fine-tuning. They stem from the need of having a physical Higgs mass consistent with m^exp_h ' 125 GeV and from the requirement of rather large tan β. Let us comment on them in order.

Fine-tuning to get m^exp_h ' 125 GeV. As is well known, the tree-level Higgs mass in the MSSM is given by (m²_h)_tree−level = M_Z²cos²2β, so radiative corrections are needed in order to reconcile it with the experimental value. A simplified expression of such correc- tions [82–84], useful for the sake of the discussion, is

δm²_h = 3G_F

√2π²m⁴_t log m²_˜t m²_t

! +X_t²

m²_˜t 1− X_t² 12m²_˜t

!!

+ · · · , (2.11)

with m_˜t the average stop mass and Xt = At− µ cot β. The Xt-contribution arises from the threshold corrections to the quartic coupling at the stop scale. This correction is maximized for Xt = √

6m˜t (Xt ' 2m˜t when higher orders are included). Notice that if the threshold correction were not present one would need heavy stops (of about 3 TeV once higher order corrections are included) for large tan β (and much heavier as tan β decreases, see ref. [85,86]); which is inconsistent with the requirements of Natural SUSY in its original formulation. However, taking Xt close to the “maximal” value, it is possible to obtain the correct Higgs mass with rather light stops, even in the 500− 700 GeV range;

a fact frequently invoked in the literature to reconcile the Higgs mass with Natural SUSY.

On the other side, requiring Xt ∼ maximal, amounts also to a certain fine-tuning if one needs to lie close to such value with great precision. The precision (and thus the fine- tuning) required depends in turn on the values of tan β and the stop masses. Therefore, when analyzing the naturalness issue one should take into account, besides the fine-tuning associated with the electroweak breaking, the one associated with the precise value required for Xt. In subsection6.1 we will discuss the size of this fine-tuning in further detail.

Fine-tuning to get large tan β. The value of tan β ≡ hHui/hHdi is given, at tree level, by

2

tan β ' sin 2β = 2Bµ m²_H

d+ m²_H

u+ 2µ² = 2Bµ

m²_A , (2.12)

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where m_A is the mass of the pseudoscalar Higgs state; all the quantities above are under- stood to be evaluated at the low-scale. Clearly, in order to get large tan β one needs small Bµ at low-energy. However, even starting with vanishing B at M_X one gets a large radia- tive correction due to the RG running. Consequently, very large values of tan β are very fine-tuned,³ as they require a cancellation between the initial value of B and the radiative contributions. On the other hand, moderately large values may be non-fined-tuned, de- pending on the size of the RG contribution to Bµ and the value of mA. Hence, a complete analysis of the MSSM naturalness has to address this potential source of fine-tuning.

3 The electroweak fine-tuning of the MSSM

In the MSSM, the vacuum expectation value of the Higgs, v²/2 = |hHui|² +|hHdi|², is given, at tree-level, by the minimization relation

−1

8(g²+ g⁰²)v² =−M_Z²

2 = µ²−m²_H

d− m²Hutan²β

tan²β− 1 . (3.1)

As is well known, the value of tan β must be rather large, so that the tree-level Higgs mass, (m²_h)_tree−level = M_Z²cos²2β, is as large as possible, ' MZ²; otherwise, the radiative corrections needed to reconcile the Higgs mass with its experimental value, would imply gigantic stop masses [85,86] (see subsection 2.3 above) and thus an extremely fine-tuned scenario. Notice here that the focus-point regime is not useful to cure such fine-tuning since it only works if tan β is rather large and stop masses are not huge.

Therefore, for Natural SUSY the limit of large tan β is the relevant one. Then, the relation (3.1) gets simplified

−1

8(g²+ g⁰²)v² =−M_Z²

2 = µ²+ m²_Hu . (3.2)

The two terms on the r.h.s. have opposite signs and their absolute values are typically much larger than M_Z², hence the potential fine-tuning associated to the electroweak breaking.

It is well-known that the radiative corrections to the Higgs potential reduce the fine- tuning [26]. This effect can be honestly included taking into account that the effective quartic coupling of the SM-like Higgs runs from its initial value at the SUSY threshold,⁴ λ(Q_threshold) = ¹₈(g²+g⁰²), until its final value at the electroweak scale, λ(QEW). The effect of this running is equivalent to include the radiative contributions to the Higgs quartic cou- pling in the effective potential, which increase the tree-level Higgs mass, (m²_h)_tree−level = 2λ(Q_threshold)v² = M_Z², up to the experimental one, m²_h = 2λ(Q_EW)v². Therefore, re- placing λ_tree−level by the radiatively-corrected quartic coupling is equivalent to replace M_Z² → m²_h in eq. (3.2) above, i.e.

−m²_h

2 = µ²+ m²_Hu, (3.3)

3The existence of this fine-tuning was first observed in ref. [87,88] and has been discussed, from the Bayesian point of view in ref. [89].

4A convenient choice of the SUSY-threshold is the average stop mass, since the 1-loop correction to the Higgs potential is dominated by the stop contribution. Hence, choosing Qthreshold ' m˜t, the 1-loop correction is minimized and the Higgs potential is well approximated by the tree-level form.

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which is the expression from which we will evaluate the electroweak fine-tuning in the MSSM. As mentioned above, the radiative corrections slightly alleviate this fine-tuning, since m_h > M_Z.

3.1 The measure of the fine-tuning

It is a common practice to quantify the amount of fine-tuning using the parametrization first proposed by Ellis et al. [90] and Barbieri and Giudice [25], which in our case reads

∂m²_h

∂θi

= ∆θi

m²_h θi

, ∆≡ Max |∆θi| , (3.4)

where θi is an independent parameter that defines the model under consideration and ∆_θi is the fine-tuning parameter associated to it. Typically θ_i are the initial (high-energy) values of the soft terms and the µ parameter. Nevertheless, for specific scenarios of SUSY breaking and transmission to the observable sector, the initial parameters might be par- ticular theoretical parameters that define the scenario and hence determine the soft terms, e.g. a Goldstino angle in scenarios of moduli-dominated SUSY breaking. We will comment further on this issue in subsection3.2.

It is worth to briefly comment on the statistical meaning of ∆_θi. In ref. [28] it was argued that (the maximum of all)|∆θi| represents the inverse of the probability of a cancel- lation among terms of a given size to obtain a result which is |∆θi| times smaller. This can be intuitively seen as follows. Expanding m²_h(θ_i) around a point in the parameter space that gives the desired cancellation, say {θi⁰}, up to first order in the parameters, one finds that only a small neighborhood δθ_i ∼ θ⁰i/∆_θi around this point gives a value of m²_h smaller or equal to the experimental value [28]. Therefore, if one assumes that θi could reason- ably have taken any value of the order of magnitude of θ_i⁰, then only for a small fraction δθ_i/θ⁰_i

∼ ∆⁻¹_θi of this region one gets m²_h <

∼ (m^exph )², hence the rough probabilistic mean- ing of ∆θi. Note that the value of ∆ can be interpreted as the inverse of the p-value to get the correct value of m²_h. If θ is the parameter that gives the maximum ∆ parameter, then⁵

p−value '    

δθ θ₀   

≡ ∆⁻¹ . (3.5)

It is noteworthy that for the previous arguments it was implicitly assumed that the possible values of a θ_i−parameter are distributed, with approximately flat probability, in the [0, θ⁰i] range. In a Bayesian language, the prior on the parameters was assumed to be flat, within the mentioned range. If the assumptions are different (either because the allowed ranges of some parameters are restricted by theoretical consistency or experimental data, or because the priors are not flat), then the probabilistic interpretation has to be consistently modified.

These issues become more transparent using a Bayesian approach.

5Notice that in the particular case when θ⁰minimizes the value of mh, then ∂mh/∂θ|_θ=θ

0= 0. This lack of sensitivity at first order when θ0 is close to an stationary point, would seemingly imply no fine-tuning, according to the “standard criterion”. However, from the above discussion, it is clear that in this case the expansion at first order is meaningless; one should start at second order, and then it becomes clear that the fine-tuning is really very high since only when θ is close to θ0, one gets m²_h<

∼ (m^exph )². In order words, the associated p-value would be very small.

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In a Bayesian analysis, the goal is to generate a map of the relative probability of the different regions of the parameter space of the model under consideration (MSSM in our case), using all the available (theoretical and experimental) information. This is the so-called posterior probability, p(θi|data), where ‘data’ stands for all the experimental information and θi represent the various parameters of the model. The posterior is given by the Bayes’ Theorem

p(θi|data) = p(data|θi) p(θi) 1

p(data), (3.6)

where p(data|θi) is the likelihood (sometimes denoted byL), i.e. the probability density of observing the given data if nature has chosen to be at the{θi} point of the parameter space (this is the quantity used in frequentist approaches); p(θi) is the prior, i.e. the “theoretical”

probability density that we assign a priori to the point in the parameter space; and, finally, p(data) is a normalization factor which plays no role unless one wishes to compare different classes of models.

For the sake of concreteness, let us focus on a particular parameter defining the MSSM, namely the µ−parameter.⁶ Now, instead of solving µ in terms of MZ and the other supersymmetric parameters using the minimization conditions (as usual), one can (actually should) treat M_Z^exp, i.e. the electroweak scale, as experimental data on a similar footing with the other observables, entering the total likelihood, L. Approximating the MZ likelihood as a Dirac delta,

p(data|M1, M₂,· · · , µ) ' δ(MZ− M_Z^exp) Lrest, (3.7) whereLrestis the likelihood associated to all the physical observables except M_Z, one can marginalize the µ−parameter

p(M₁, M₂,· · · | data) = Z

dµ p(M₁, M₂,· · · , µ|data)

∝ Lrest

dµ dM_Z

   µZ

p(M₁, M₂,· · · , µZ) , (3.8) where we have used eqs. (3.6), (3.7). Here µ_Z is the value of µ that reproduces M_Z^expfor the given values of {M1, M₂,· · · }, and p(M1, M₂,· · · , µ) is the prior in the initial parameters (still undefined). Note that the above Jacobian factor in eq. (3.8) can be written as⁷

dµ dMZ

   µZ

∝    

µ

∆µ

   µ_Z

, (3.9)

where the constant factors are absorbed in the global normalization factor of eq. (3.6). The important point is that the relative probability density of a point in the MSSM parameter

6Of course, one can take here another parameter and the argument goes the same (actually, in some theoretical scenarios µ may be not an initial parameter). On the other hand, µ is a convenient choice since it is the parameter usually solved in terms of MZ in phenomenological analyses.

7Notice that the dependence of MZ on µ is through eq. (3.3), which determines the Higgs VEV. Thus

dM_Z²

dµ ∝^dm_dµ^2h.

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space is multiplied by ∆⁻¹_µ , which is consistent with the above probabilistic interpretation of ∆ [45, 89, 91, 92]. Actually, the equivalence is exact if one assumes that the prior in the parameters is factorizable, i.e. p(M₁, M₂,· · · , µ) = p(M1)p(M₂)· · · p(µ), and p(µZ)∝ 1/µZ, so that the numerator in the r.h.s. of (3.9) is canceled when plugged in eq. (3.8).

This assumption can be realized in two different ways. First, if µ has a flat prior with range

∼ [0, µZ], then the normalization of the µ−prior goes like ∝ 1/µZ. This is exactly the kind of implicit assumption discussed above. Alternatively, if µ has a logarithmically flat prior, then p(µ)∝ 1/µ, with the same result (this is probably the most sensible prior to adopt since it means that all magnitudes of the SUSY parameters are equally probable).

In summary, the standard measure of the fine-tuning (3.4) is reasonable and can be rigorously justified using Bayesian methods. In consequence, we will use it throughout the paper. Nevertheless, it should be kept in mind that the previous Bayesian analysis also provides the implicit assumptions for its validity. If a particular theoretical model does not fulfill them, the standard criterion is inappropriate and should be consistently modified.

3.2 Generic expression for the fine-tuning

Clearly, in order to use the standard measure of the fine-tuning (3.4) it is necessary to write the r.h.s. of the minimization equation (3.3) in terms of the initial parameters. This in turn implies to write the low-energy values of m²_Huand µ in terms of the initial, high-energy, soft- terms and µ−term (for specific SUSY constructions, these parameters should themselves be expressed in terms of the genuine initial parameters of the model). Low-energy (LE) and high-energy (HE) parameters are related by the RG equations, which normally have to be integrated numerically. However, it is extremely convenient to express this dependence in an exact, analytical way. Fortunately, this can be straightforwardly done, since the dimensional and analytical consistency dictates the form of the dependence,

m²_Hu(LE) = c_M²

3M₃²+ c_M²

2M₂²+ c_M²

1M₁²+ c_A²

tA²_t+ c_AtM3A_tM₃+ c_M3M2M₃M₂+· · ·

· · · + cm²_Hum²_Hu+ c_m²

Q3m²_Q3 + c_m²

U3m²_U3+· · · (3.10)

µ(LE) = cµµ , (3.11)

where Mi are the SU(3)× SU(2) × U(1)^Y gaugino masses, At is the top trilinear scalar coupling; and m_Hu, m_Q3, m_U3 are the masses of the H_u−Higgs, the third-generation squark doublet and the stop singlet respectively, all of them understood at the HE scale. The numerical coefficients, c_M²

3, c_M²

2, . . . are obtained by fitting the result of the numerical integration of the RGEs to eqs. (3.10), (3.11), a task that we perform carefully in the subsection3.3.

The above equations (3.10), (3.11) replace the one-loop LL expressions (2.4), (2.6) used in the standard Natural-SUSY treatment. If one considers the initial values of the soft parameters and µ as the independent parameters that define the MSSM, then one can easily extract the associated fine-tuning by applying eq. (3.4) to (3.3), and replacing m²_H

u

by the expression (3.10). Note that the above definition of ∆, eq. (3.4), is actually not very different from the definition (2.2) used in ref. [76]; actually they are identical for the parameters that enter as a single term in the sum of eq. (3.10), e.g. m²_˜

U3. Nevertheless,

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eq. (3.4) differs from eq. (2.2) when the parameter enters in several terms, e.g. M₃.⁸ On the other hand, the definition (3.4), besides being statistically more meaningful, allows to study scenarios where the initial parameters are not soft masses.

From eqs. (3.3), (3.10), (3.11)) it is easy to derive the ∆−parameters (3.4) for any MSSM scenario. A common practice is to consider the (HE) soft terms and the µ−term as the independent parameters, say

Θα =µ, M₃, M₂, M₁, At, m²_Hu, m²_Hd, m²_U3, m²_Q3,· · · , (3.12) which is equivalent to the so-called “Unconstrained MSSM”.⁹ Then one easily com- putes ∆_Θα

∆_Θα = Θ_α m²_h

∂m²_h

∂Θ_α =−2Θ_α m²_h

∂m²_Hu

∂Θ_α . (3.13)

E.g. ∆_M3 is given by

∆M3 =−2M3

m²_h

 2c_M²

3M3+ cAtM3At+ cM3M2M2+· · ·

. (3.14)

The identification ^∂m_∂Θ^2h

α ' −2^∂m

2 Hu

∂Θα in eq. (3.13) comes from eq. (3.3) and thus is valid for all the parameters except µ, for which we simply have

∆µ= µ m²_h

∂m²_h

∂µ =−4c²µ

µ²

m²_h=−4 µ(LE) m²_h

2

. (3.15)

Besides, the term proportional to m²_H

d in eq. (3.1), which was subsequently neglected, can give relatively important contributions to ∆_m²

Hd if tan β is not too large (^<∼10), namely

∆_m²

Hd ' −2m²_H

d

m²_h

 c_m²

Hd − c⁰_m²

Hd (tan²β− 1)⁻¹



, (3.16)

where c⁰_m2

Hd ' 1 denotes the c−coefficient of m²Hd in the expression of the LE value of m²_H

d

itself, see table 3 in appendix. In any case, the contribution of m²_H

d to the fine-tuning is always marginal.

Note that for any other theoretical scenario, the ∆s associated with the genuine initial parameters, say θ_i, can be written in terms of ∆_Θα using the chain rule

∆_θi ≡ ∂ ln m²_h

∂ ln θi

=X

α

∆_Θα∂ ln Θ_α

∂ ln θi

= θ_i m²_h

X

α

∂m²_h

∂Θα

∂Θ_α

∂θi

. (3.17)

8Indeed, if eq. (2.2) was refined to incorporate the M3−dependent contributions to m², e.g. through their impact in the stop mixing, the result would be very similar to that of eq. (3.4).

9The name “Unconstrained MSSM” could be a bit misleading in this context, since it would seem to imply that one is not doing any assumptions about the soft terms. But there is in fact an assumption, namely that they are not correlated. Note in particular that although the parameter space of the Unconstrained MSSM includes any MSSM, e.g. the “Constrained MSSM”, the calculation of the fine-tuning for the latter requires to take into account a specific correlation between various soft-terms. Still, we are showing in this section that the results for the Unconstrained MSSM allow to easily evaluate the fine-tuning in any other MSSM scenario.

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Finally, in order to obtain fine-tuning bounds on the parameters of the model we demand

|∆θi| ^<∼ ∆^max, where ∆^max is the maximum amount of fine-tuning one is willing to ac- cept. E.g.

∆^max= 100 , (3.18)

represents a fine-tuning of ∼ 1%.

3.3 The fit to the low-energy quantities

Fits of the kind of eq. (3.10) can be found in the literature, see e.g. [93, 94]. However, though useful, they should be refined in several ways in order to perform a precise fine- tuning analysis. The most important improvement is a careful treatment of the various threshold scales. In particular, the initial MSSM parameters (i.e. the soft terms and the µ−parameter) are defined at a high-energy (HE) scale, which is usually identified as MX, i.e. the scale at which the gauge couplings unify. Although this is a reasonable assumption, it is convenient to consider the HE scale as an unknown; e.g. in gauge-mediated scenarios it can be in principle any scale. The low-energy (LE) scale at which one sets the SUSY threshold and the supersymmetric spectrum is computed, is also model-dependent. A reasonable choice is to take M_LE as the averaged stop masses. As discussed above, at this scale the 1-loop corrections to the effective potential are minimized, so that the potential is well approximated by the tree-level expression; thus eq. (3.10) should be understood at this scale. Nevertheless, in many fits from the literature M_LE is identified with MZ. Finally, some parameters are inputs at M_Z, e.g. the gauge couplings, while others, like the soft B−parameter (the coefficient of the bilinear Higgs coupling), have to be evaluated in order to reproduce the correct electroweak breaking with the value of tan β chosen. Similarly, the value of the top Yukawa-coupling has to be settled at high energy in such a way that it reproduces the value of the top mass at the electroweak scale (which is below the LE scale). All this requires to divide the RG-running into two segments, [M_EW, M_LE] and [M_LE, M_HE]. Besides this refinement, we have integrated the RG-equations at two-loop order, using SARAH 4.1.0 [95].

The results of the fits for all the LE quantities for tan β = 10 and M_HE = MX are given in appendix, tables3,4,5,6,7,8, and9. The value quoted for each c−coefficient has been evaluated at M_LE = 1 TeV. The dependence of the c−coefficients on MLE is logarithmic and can be well approximated by

ci(MLE)' ci(1 TeV) + biln MLE

1 TeV . (3.19)

The value of the b_i coefficients is also given in tables 3–9 (for M_HE = M_X). Certainly, the value of MLE ∼ m˜t is itself a (complicated) function of the initial soft parameters.

Nevertheless, it is typically dominated by the (RG) gluino contribution, M_LE ∼ m˜t ∼

√3|M3| for MHE = M_X. This represents an additional dependence of m²_H

u on M₃, which should be taken into account when computing ∆M3. Actually, this effect diminishes the fine-tuning associated to M₃ (which is among the most important ones) because the impact of an increase of M₃ in the value of m²_H

u becomes (slightly) compensated by the increase

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of the LE scale and the consequent decrease of the c_M2

3 coefficient in eq. (3.10). We have incorporated this fact in the computations of the fine-tuning.

Let us now turn to the dependence of the fine-tuning on the high-energy scale, M_HE. The absolute values of all the c−coefficients in the fits decrease with MHE, except perhaps the coefficient that multiplies the parameter under consideration (e.g. c_m²

Hu in eq. (3.10)).

In the limit M_HE → MLE the latter becomes 1, and the others go to zero. Obviously, the fine-tuning decreases as MHE decreases. The actual dependence of the c−coefficients on M_HEhas to do with the loop-order at which it arises. If it does at one-loop, the dependence is logarithmic-like, e.g. for c_M²

2 in eq. (3.10); if it does at two-loop, the dependence goes like ∼ (log MHE)², e.g. for c_M²

3. These dependences are shown in figures4,5,6,7 and8.

In summary, with the help of tables 3–9and Figs 4–8 it is straightforward to evaluate the fine-tuning parameters of any MSSM scenario.

4 The naturalness bounds

4.1 Bounds on the initial (high-energy) parameters

Let us explore further the size and structure of the fine-tuning, and the corre- sponding bounds on the initial parameters, in the unconstrained MSSM, i.e. tak- ing as initial parameters the HE values of the soft terms and the µ-term: Θ_α =

µ, M3, M₂, M₁, A_t, m²_H

u, m²_H

d, m²_U

3, m²_Q

3,· · · . This is interesting by itself, and, as dis- cussed above, it can be considered as the first step to compute the fine-tuning in any theoretical scenario. For any of those parameters we demand

|∆Θα|^∼<∆^max, (4.1)

where ∆_Θα are given by eq. (3.13). Now, for the parameters that appear just once in eqs. (3.10), (3.11) the corresponding naturalness bound (4.1) is trivial and has the form of an upper limit on the parameter size. For dimensional reasons this is exactly the case for dimension-two parameters in mass units, e.g. for the squared stop masses

  ∆m²_Q3

   =

−2m²_Q3 m²_h c_m2

Q3

<∼∆^max −→ m²_Q3 ^<∼1.36 ∆^max m²_h (4.2) 

 ∆m²_U3

   =

−2m²_U3 m²_h c_m²

U3

∼<∆^max −→ m²_˜t

R

∼<1.72 ∆^max m²_h, (4.3) where we have plugged , cm_Q3 =−0.367, cm_U3 =−0.29, which correspond to MHE = MX

and M_LE= 1 TeV, see table3. For ∆^max= 100, we get mQ3 <

∼1.46 TeV, mU3 <

∼ 1.64 TeV, substantially higher than the usual quoted bounds [77]. This is mainly due to the refined RG analysis and the use of the radiatively upgraded expression eq. (3.3), rather than eq. (3.2), to evaluate the fine-tuning. We stress that these are the bounds on the high- energy soft masses, the bounds on the physical masses will be worked out in subsection4.3.

The naturalness bounds for the other (HE) dimension-two parameters (m²_D3, m²_Q1,2, m²_U1,2, m²_D

1,2, m²_L

3, . . . ) have a form similar to eqs. (4.2), (4.3) and are also higher than usually quoted. Due to its peculiar RGE, this is also the case of the µ−parameter, see eq. (3.15).

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On the other hand, for dimension-one parameters (except µ) the naturalness bounds (4.1) appear mixed. In particular, this is the case for the bounds associated to M₃, M₂, A_t. From eqs. (3.13) and (3.10)

|∆M3| = 1 m²_h

6.41M₃²− 0.57AtM₃+ 0.27M₃M₂

_∼^<∆^max (4.4)

|∆M2| = 1 m²_h

−0.81M2²− 0.14AtM₂+ 0.27M₃M₂

^<_∼∆^max (4.5)

|∆At| = 1 m²_h

0.44A²_t − 0.57AtM₃− 0.14AtM₂

_∼^<∆^max, (4.6) where, again, we have plugged the values of the c−coefficients corresponding to MHE = M_X and M_LE= 1 TeV. Other parameters, like M₁, A_b, get also mixed with them in the bounds, but their coefficients are much smaller, so we have neglected them. We show in figure 1 the region in the {M2, M₃, A_t} space that fulfills the inequalities for ∆^max = 100. The figure is close to a prism. Their faces are given by the following approximate solution to eqs. (4.4)–(4.6)

M₃^max ' ± mh

r ∆^max 6.41 + 1

12.82(0.57At− 0.27M2) (4.7) M₂^max ' ± mh

r ∆^max 0.81 + 1

1.62(0.27M₃− 0.14At) (4.8) A^max_t ' ± mh

r ∆^max 0.44 + 1

0.88(0.57M3+ 0.14M2) , (4.9) where the superscript “max” denotes the, positive and negative, values of the parame- ter that saturate inequalities (4.4)–(4.6). Thus eqs. (4.7)–(4.9) represent the naturalness bounds to M₃, M₂, At. Each individual bound depends on the values of the other param- eters due to the presence of the mixed terms. Depending on the relative signs of the soft terms, the bounds can be larger or smaller than those obtained when neglecting the mixed terms. However, the presence of the latter stretches each individual absolute upper bound in a non-negligible way, by doing an appropriate choice of the other soft terms (compatible with their own fine-tuning condition).

A generic, approximate, expression for the absolute upper bound on a dimension-one parameter, i.e. Mi (Mi = M₃, M₂, M₁, A_t, A_b. . .) can be obtained by replacing the other dimension-one parameters,Mj6=i, by the values that saturate their zeroth-order fine-tuning bounds,±M^maxj ' mh

q∆^max/4|cM²_j|, with the appropriate sign; namely

|Mi| < m_h 2

s∆^max

|cM²_i|



1 +X

j6=i

1 4

|c^MiM_j| q|cM²_ic_M²

j|



 . (4.10)

In practice, in order to obtain the absolute upper bounds on M₃, M₂, A_t we have ignored the presence of additional parameters (M1, Ab,· · · ) in (4.10). Its inclusion would stretch even further the absolute bounds, but quite slightly and artificially since this would imply a certain conspiracy between soft parameters. As a matter of fact, even playing just

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Figure 1. Region in the{M², M3, At} space that fulfills eqs. (4.4)–(4.6) for ∆^max = 100 (axes units: TeVs). For other values, ap∆^max/100 scaling factor has to be applied.

with the three parameters which show a sizeable correlation, i.e. {M3, M₂, At}, implies a certain degree of conspiracy to get the maximum value quoted in (4.10). This means that the bound (4.10) is conservative. A more restrictive and rigorous bound can be obtained by demanding that the addition in quadrature of the ∆iparameters never exceeds the reference value, ∆^max. In any case, since the fine-tuning conditions of M₃, M₂, A_t are correlated, as shown in eqs. (4.4)–(4.6), the most meaningful approach is to determine the regions of the parameter space simultaneously consistent with all the fine-tuning conditions. This will be done in detail in section 6.2 below. The numerical modification of eqs. (4.4)–(4.9) for different values of M_LE, M_HE can be straightforwardly obtained from table 3and figure4.

Choosing ∆^max= 100, eqs. (4.7)–(4.9) give |M3|^∼<610 GeV, |M2|^∼<1630 GeV, |A^t|^∼<

2430 GeV. The limit on M₃ is similar to the one found by Feng [77], although this is in part a coincidence. In ref. [77] it was chosen M₃², rather than M₃, as an independent parameter; which reduces the associated ∆_M3 by a factor of 2. So, their bound on M₃ was increased (quite artificially in our opinion) by √

2. On the other hand, in ref. [77] the RG running was not done in two steps, but simply running all the way from MX till MZ. Furthermore, they did not consider the mixed terms of eq. (4.4). And finally they used eq. (3.2) instead of eq. (3.3) to evaluate the fine-tuning. It turns out that, all together, these three approximations increase the estimate of the fine tuning, thus decreasing the upper bound on M₃ by a factor which happens to be ∼ 1/√

2.

Actually, for the particular case of the M3−parameter this is not the end of the story.

As discussed in subsection 3.2, the c_M²

3 coefficient has a dependence on M_LEapproximately given by eq. (3.19). Since M_LE' m˜tand typically m²_˜t ' ¹₂(c^(Q_M2³⁾

3

+ c^(U_M³2⁾ 3

)²M₃², where c^(Q_M2³⁾ 3

, c^(U_M³2⁾

3

are the coefficients of M₃² in the LE expression of m²_Q

3, m²_U

3 (given in table 4 and